Introduce collapsible proofs to some articles

1 files changed, 24 insertions, 17 deletions

diff --git a/content/know/concept/dirac-delta-function/index.pdc b/content/know/concept/dirac-delta-function/index.pdc
index 76b6e97..9eecefd 100644
--- a/content/know/concept/dirac-delta-function/index.pdc
+++ b/content/know/concept/dirac-delta-function/index.pdc
@@ -21,7 +21,7 @@ defined to be 1:
 
 $$\begin{aligned}
     \boxed{
-        \delta(x) =
+        \delta(x) \equiv
         \begin{cases}
             +\infty & \mathrm{if}\: x = 0 \\
             0 & \mathrm{if}\: x \neq 0
@@ -56,12 +56,10 @@ following integral, which appears very often in the context of
 [Fourier transforms](/know/concept/fourier-transform/):
 
 $$\begin{aligned}
-    \boxed{
-        \delta(x)
-        %= \lim_{n \to +\infty} \!\Big\{\frac{\sin(n x)}{\pi x}\Big\}
-        = \frac{1}{2\pi} \int_{-\infty}^\infty \exp(i k x) \dd{k}
-        \:\:\propto\:\: \hat{\mathcal{F}}\{1\}
-    }
+    \delta(x)
+    = \lim_{n \to +\infty} \!\Big\{\frac{\sin(n x)}{\pi x}\Big\}
+    = \frac{1}{2\pi} \int_{-\infty}^\infty \exp(i k x) \dd{k}
+    \:\:\propto\:\: \hat{\mathcal{F}}\{1\}
 \end{aligned}$$
 
 When the argument of $\delta(x)$ is scaled, the delta function is itself scaled:
@@ -72,18 +70,22 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-*__Proof.__ Because it is symmetric, $\delta(s x) = \delta(|s| x)$. Then by
-substituting $\sigma = |s| x$:*
+<div class="accordion">
+<input type="checkbox" id="proof-scale"/>
+<label for="proof-scale">Proof</label>
+<div class="hidden">
+<label for="proof-scale">Proof.</label>
+Because it is symmetric, $\delta(s x) = \delta(|s| x)$.
+Then by substituting $\sigma = |s| x$:
 
 $$\begin{aligned}
     \int \delta(|s| x) \dd{x}
     &= \frac{1}{|s|} \int \delta(\sigma) \dd{\sigma} = \frac{1}{|s|}
 \end{aligned}$$
+</div>
+</div>
 
-*__Q.E.D.__*
-
-An even more impressive property is the behaviour of the derivative of
-$\delta(x)$:
+An even more impressive property is the behaviour of the derivative of $\delta(x)$:
 
 $$\begin{aligned}
     \boxed{
@@ -91,16 +93,21 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-*__Proof.__ Note which variable is used for the
-differentiation, and that $\delta'(x - \xi) = - \delta'(\xi - x)$:*
+<div class="accordion">
+<input type="checkbox" id="proof-dv1"/>
+<label for="proof-dv1">Proof</label>
+<div class="hidden">
+<label for="proof-dv1">Proof.</label>
+Note which variable is used for the
+differentiation, and that $\delta'(x - \xi) = - \delta'(\xi - x)$:
 
 $$\begin{aligned}
     \int f(\xi) \: \dv{\delta(x - \xi)}{x} \dd{\xi}
     &= \dv{x} \int f(\xi) \: \delta(x - \xi) \dd{x}
     = f'(x)
 \end{aligned}$$
-
-*__Q.E.D.__*
+</div>
+</div>
 
 This property also generalizes nicely for the higher-order derivatives: